Then, the \(X^2\) Pearson statistic is
\[X^2=\sum_{}^{}\frac{(N_{ij}-n^*_{ij})^2}{n^*_{ij}}\underset{H_0}{\stackrel{\cdot}\sim} \chi^2_{(s-1)(t-1)}\]
Then, using the observed frequencies, \(n_{ij}\), and the expected frequencies, \(n^*_{ij}\), we can obtain the observed test statistics
\[t_0=\sum_{}^{}\frac{(n_{ij}-n^*_{ij})^2}{n^*_{ij}}\]
By fixing the significance level \(\alpha\), \(\text{we reject} \, \, {\rm H_0}\,\, \text{if}\,\, t_0>\chi^2_{(s-1)(t-1); 1-\alpha}\) or equivalently if \[t_0 \in \mathcal{R}_\alpha=(\chi^2_{(s-1)(t-1); 1-\alpha},+\infty) \]